If Supergirl kisses someone can she erase their memory of things like Superman can?
The power you mention, Superman's ability to erase memories, was one of many powers that Superman gained over his lifetime as his power set spun out of control. In order to keep him interesting, writers would keep adding new powers as needed. The memory manipulation ability is an early "silver age" (1950-1980s) era Superman power (thanks @Valorum) that was not used very often, one of many hypnotic abilities Superman had in that time period. Since this is the Superman most people were familiar with when Donner's movies came out, those are the powers Donner used for his character.
In the current incarnation of Superman, however, most of the more extreme powers have been removed, and the current Superman has never shown the memory erasing ability. In addition, no incarnation of Supergirl that I'm aware of has shown that power herself.
So, unless the TV show chooses to introduce a new power (which they can easily do), Kara's kiss probably cannot erase someone's memory.